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cdtools/CDTools/tools/analysis/analysis.py
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"""Contains basic functions for analyzing the results of reconstructions
The functions in this module are designed to work either with pytorch tensors
or numpy arrays, so they can be used either directly after reconstructions
on the attributes of the models themselves, or after-the-fact once the
data has been stored in numpy arrays.
"""
import torch as t
import numpy as np
from CDTools.tools import image_processing as ip
from scipy import fftpack
from scipy import linalg as sla
from scipy import special
__all__ = ['orthogonalize_probes', 'standardize', 'synthesize_reconstructions',
'calc_consistency_prtf', 'calc_deconvolved_cross_correlation',
'calc_frc', 'calc_vn_entropy', 'calc_top_mode_fraction',
'calc_rms_error', 'calc_fidelity', 'calc_generalized_rms_error']
def orthogonalize_probes(probes, density_matrix=None, keep_transform=False, normalize=False):
"""Orthogonalizes a set of incoherently mixing probes
The strategy is to define a reduced orthogonal basis that spans
all of the retrieved probes, and then build the density matrix
defined by the probes in that basis. After diagonalization, the
eigenvectors can be recast into the original basis and returned
By default, it assumes that the set of probes are defined just as
standard incoherently mixing probe modes, and orthogonalizes them.
However, if a density matrix is explicitly given, it will instead
consider the problem of extracting the eigenbasis of the matrix
probes * denstity_matrix * probes^dagger, where probes is the
column matrix of the given probe functions. This latter problem arises
in the generalization of the probe mixing model, and reduces to the
simpler case when the density matrix is equal to the identity matrix
If the parameter "keep_transform" is set, the function will additionally
return the matrix A such that A * ortho_probes^dagger = probes^dagger
If the parameter "normalize" is False (as is the default), the variation
in intensities in the probe modes will be kept in the probe modes, as is
natural for a purely incoherent model. If it is set to "True", the
returned probe modes will all be normalized instead.
Parameters
----------
probes : array
An l x n x m complex array representing a stack of probes
density_matrix : np.array
An optional l x l density matrix further elaborating on the state
keep_transform : bool
Default False, whether to return the map from probes to ortho_probes
normalize : bool
Default False, whether to normalize the probe modes
Returns
-------
ortho_probes: array
An l x n x m complex array representing a stack of probes
"""
try:
probes = probes.detach().cpu().numpy()
send_to_torch = True
except:
send_to_torch = False
# We can do the orthogonalization with an SVD, so first we have to
# reshape the final two dimensions (the image shape) into a single
# vectorized dimension. This matrix is probes^dagger, hence the
# conjugation
probes_mat = probes.reshape(probes.shape[0],
probes.shape[1]*probes.shape[2])
if density_matrix is None:
density_matrix = np.eye(probes.shape[0])
# next we want to extract the eigendecomposition of the density matrix
# itself
w,v = sla.eigh(density_matrix)
w = w[::-1]
v = v[:,::-1]
# We do this just to avoid total failure when the density
# matrix is not positive definite.
# In most cases (such as when rho is generated directly from some other
# matrix A such that rho=A A^dagger), w should never have any negative
# entries.
w = np.maximum(w,0)
B_dagger = np.dot(np.diag(np.sqrt(w)), v.conj().transpose())
#u,s,vh = np.linalg.svd(np.dot(B_dagger,probes_mat), full_matrices=False)
u,s,vh = sla.svd(np.dot(B_dagger,probes_mat), full_matrices=False)
if normalize:
ortho_probes = vh.reshape(probes.shape[0],
probes.shape[1],
probes.shape[2])
B_dagger_inv = np.linalg.pinv(B_dagger)
A = np.dot(B_dagger_inv,np.dot(u,np.diag(s)))
#A_dagger = np.dot(np.linalg.pinv(np.diag(s)),
# np.dot(np.transpose(u).conj(),B_dagger))
else:
ortho_probes = np.dot(np.diag(s),vh).reshape(probes.shape[0],
probes.shape[1],
probes.shape[2])
B_dagger_inv = np.linalg.pinv(B_dagger)
A = np.dot(B_dagger_inv,u)
#A_dagger = np.dot(np.transpose(u).conj(),B_dagger)
if send_to_torch:
ortho_probes = t.as_tensor(np.stack(ortho_probes))
A = t.as_tensor(A)
if keep_transform:
return ortho_probes, A#_dagger
else:
return ortho_probes
def standardize(probe, obj, obj_slice=None, correct_ramp=False):
"""Standardizes a probe and object to prepare them for comparison
There are a number of ambiguities in the definition of a ptychographic
reconstruction. This function makes an explicit choice for each ambiguity
to allow comparisons between independent reconstructions without confusing
these ambiguities for real differences between the reconstructions.
The ambiguities and standardizations are:
1) a. Probe and object can be scaled inversely to one another
b. So we set the probe intensity to an average per-pixel value of 1
2) a. The probe and object can aquire equal and opposite phase ramps
b. So we set the centroid of the FFT of the probe to zero frequency
3) a. The probe and object can each acquire an arbitrary overall phase
b. So we set the phase of the sum of all values of both the probe and object to 0
When dealing with the properties of the object, a slice is used by
default as the edges of the object often are dominated by unphysical
noise. The default slice is from 3/8 to 5/8 of the way across. If the
probe is actually a stack of incoherently mixing probes, then the
dominant probe mode (assumed to be the first in the list) is used, but
all the probes are updated with the same factors.
Parameters
----------
probe : array
A complex array storing a retrieved probe or stack of incoherently mixed probes
obj : array
A complex array storing a retrieved object
obj_slice : slice
Optional, a slice to take from the object for calculating normalizations
correct_ramp : bool
Default False, whether to correct for the relative phase ramps
Returns
-------
standardized_probe : array
The standardized probe
standardized_obj : array
The standardized object
"""
# First, we normalize the probe intensity to a fixed value.
probe_np = False
if isinstance(probe, np.ndarray):
probe = t.as_tensor(probe, dtype=t.complex64)
probe_np = True
obj_np = False
if isinstance(obj, np.ndarray):
obj = t.as_tensor(obj,dtype=t.complex64)
obj_np = True
# If this is a single probe and not a stack of probes
if len(probe.shape) == 2:
probe = probe[None,...]
single_probe = True
else:
single_probe = False
normalization = t.sqrt(t.mean(t.abs(probe[0])**2))
probe = probe / normalization
obj = obj * normalization
# Default slice of the object to use for alignment, etc.
if obj_slice is None:
obj_slice = np.s_[(obj.shape[0]//8)*3:(obj.shape[0]//8)*5,
(obj.shape[1]//8)*3:(obj.shape[1]//8)*5]
if correct_ramp:
# Need to check if this is actually working and, if not, why not
center_freq = ip.centroid(t.abs(t.fft.fftshift(t.fft.fft2(probe[0]),
dim=(-1,-2)))**2)
center_freq -= t.div(t.tensor(probe[0].shape,dtype=t.float32),2,rounding_mode='floor')
center_freq /= t.as_tensor(probe[0].shape,dtype=t.float32)
Is, Js = np.mgrid[:probe[0].shape[0],:probe[0].shape[1]]
probe_phase_ramp = t.exp(2j * np.pi *
(center_freq[0] * t.tensor(Is).to(t.float32) +
center_freq[1] * t.tensor(Js).to(t.float32)))
probe = probe * t.conj(probe_phase_ramp)
Is, Js = np.mgrid[:obj.shape[0],:obj.shape[1]]
obj_phase_ramp = t.exp(2j*np.pi *
(center_freq[0] * t.tensor(Is).to(t.float32) +
center_freq[1] * t.tensor(Js).to(t.float32)))
obj = obj * obj_phase_ramp
# Then, we set them to consistent absolute phases
obj_angle = t.angle(t.sum(obj[obj_slice]))
obj = obj * t.exp(-1j*obj_angle)
for i in range(probe.shape[0]):
probe_angle = t.angle(t.sum(probe[i]))
probe[i] = probe[i] * t.exp(-1j*probe_angle)
if single_probe:
probe = probe[0]
if probe_np:
probe = probe.detach().cpu().numpy()
if obj_np:
obj = obj.detach().cpu().numpy()
return probe, obj
def synthesize_reconstructions(probes, objects, use_probe=False, obj_slice=None, correct_ramp=False):
"""Takes a collection of reconstructions and outputs a single synthesized probe and object
The function first standardizes the sets of probes and objects using the
standardize function, passing through the relevant options. Then it
calculates the closest overlap of subsequent frames to subpixel
precision and uses a sinc interpolation to shift all the probes and objects
to a common frame. Then the images are summed.
Parameters
----------
probes : list(array)
A list of probes or stacks of probe modes
objects : list(array)
A list of objects
use_probe : bool
Default False, whether to use the probe or object for alignment
obj_slice : slice
Optional, A slice of the object to use for alignment and normalization
correct_ramp : bool
Default False, whether to correct for a relative phase ramp in the probe and object
Returns
-------
synth_probe : array
The synthesized probe
synth_obj : array
The synthesized object
obj_stack : list(array)
A list of standardized objects, for further processing
"""
# This should be cleaned up so it accepts anything array_like
probe_np = False
if isinstance(probes[0], np.ndarray):
probes = [t.as_tensor(probe,dtype=t.complex64) for probe in probes]
probe_np = True
obj_np = False
if isinstance(objects[0], np.ndarray):
objects = [t.as_tensor(obj,dtype=t.complex64) for obj in objects]
obj_np = True
obj_shape = np.min(np.array([obj.shape for obj in objects]),axis=0)
objects = [obj[:obj_shape[0],:obj_shape[1]] for obj in objects]
if obj_slice is None:
obj_slice = np.s_[(objects[0].shape[0]//8)*3:(objects[0].shape[0]//8)*5,
(objects[0].shape[1]//8)*3:(objects[0].shape[1]//8)*5]
synth_probe, synth_obj = standardize(probes[0].clone(), objects[0].clone(), obj_slice=obj_slice,correct_ramp=correct_ramp)
obj_stack = [synth_obj]
for i, (probe, obj) in enumerate(zip(probes[1:],objects[1:])):
probe, obj = standardize(probe.clone(), obj.clone(), obj_slice=obj_slice,correct_ramp=correct_ramp)
if use_probe:
shift = ip.find_shift(synth_probe[0],probe[0], resolution=50)
else:
shift = ip.find_shift(synth_obj[obj_slice],obj[obj_slice], resolution=50)
obj = ip.sinc_subpixel_shift(obj,np.array(shift))
if len(probe.shape) == 3:
probe = t.stack([ip.sinc_subpixel_shift(p,tuple(shift))
for p in probe],dim=0)
else:
probe = ip.sinc_subpixel_shift(probe,tuple(shift))
synth_probe = synth_probe + probe
synth_obj = synth_obj + obj
obj_stack.append(obj)
# If there only was one image
try:
i
except:
i = -1
if probe_np:
synth_probe = synth_probe.numpy()
if obj_np:
synth_obj = synth_obj.numpy()
obj_stack = [obj.numpy() for obj in obj_stack]
return synth_probe/(i+2), synth_obj/(i+2), obj_stack
def calc_consistency_prtf(synth_obj, objects, basis, obj_slice=None,nbins=None):
"""Calculates a PRTF between each the individual objects and a synthesized one
The consistency PRTF at any given spatial frequency is defined as the ratio
between the intensity of any given reconstruction and the intensity
of a synthesized or averaged reconstruction at that spatial frequency.
Typically, the PRTF is averaged over spatial frequencies with the same
magnitude.
Parameters
----------
synth_obj : array
The synthesized object in the numerator of the PRTF
objects : list(array)
A list of objects or diffraction patterns for the denomenator of the PRTF
basis : array
The basis for the reconstruction array to allow output in physical unit
obj_slice : slice
Optional, a slice of the objects to use for calculating the PRTF
nbins : int
Optional, number of bins to use in the histogram. Defaults to a sensible value
Returns
-------
freqs : array
The frequencies for the PRTF
PRTF : array
The values of the PRTF
"""
obj_np = False
if isinstance(objects[0], np.ndarray):
objects = [t.as_tensor(obj, dtype=t.complex64) for obj in objects]
obj_np = True
if isinstance(synth_obj, np.ndarray):
synth_obj = t.as_tensor(synth_obj, dtype=t.complex64)
if isinstance(basis, t.Tensor):
basis = basis.detach().cpu().numpy()
if obj_slice is None:
obj_slice = np.s_[(objects[0].shape[0]//8)*3:(objects[0].shape[0]//8)*5,
(objects[0].shape[1]//8)*3:(objects[0].shape[1]//8)*5]
if nbins is None:
nbins = np.max(synth_obj[obj_slice].shape) // 4
synth_fft = (t.abs(t.fft.fftshift(t.fft.fft2(synth_obj[obj_slice]), dim=(-1,-2)))**2).numpy()
di = np.linalg.norm(basis[:,0])
dj = np.linalg.norm(basis[:,1])
i_freqs = fftpack.fftshift(fftpack.fftfreq(synth_fft.shape[0],d=di))
j_freqs = fftpack.fftshift(fftpack.fftfreq(synth_fft.shape[1],d=dj))
Js,Is = np.meshgrid(j_freqs,i_freqs)
Rs = np.sqrt(Is**2+Js**2)
synth_ints, bins = np.histogram(Rs,bins=nbins,weights=synth_fft)
prtfs = []
for obj in objects:
obj = obj[obj_slice]
single_fft = (t.abs(t.fft.fftshift(t.fft.fft2(obj),
dim=(-1,-2)))**2).numpy()
single_ints, bins = np.histogram(Rs,bins=nbins,weights=single_fft)
prtfs.append(synth_ints/single_ints)
prtf = np.mean(prtfs,axis=0)
if not obj_np:
bins = t.Tensor(bins)
prtf = t.Tensor(prtf)
return bins[:-1], prtf
def calc_deconvolved_cross_correlation(im1, im2, im_slice=None):
"""Calculates a cross-correlation between two images with their autocorrelations deconvolved.
This is formally defined as the inverse Fourier transform of the normalized
product of the Fourier transforms of the two images. It results in a
kernel, whose characteristic size is related to the exactness of the
possible alignment between the two images, on top of a random background
Parameters
----------
im1 : array
The first image, as a complex or real valued array
im2 : array
The first image, as a complex or real valued array
im_slice : slice
Default is from 3/8 to 5/8 across the image, a slice to use in the processing.
Returns
-------
corr : array
The complex-valued deconvolved cross-correlation, in real space
"""
im_np = False
if isinstance(im1, np.ndarray):
im1 = t.as_tensor(im1)
im_np = True
if isinstance(im2, np.ndarray):
im2 = t.as_tensor(im2)
im_np = True
if im_slice is None:
im_slice = np.s_[(im1.shape[0]//8)*3:(im1.shape[0]//8)*5,
(im1.shape[1]//8)*3:(im1.shape[1]//8)*5]
cor_fft = t.fft.fft2(im1[im_slice]) * \
t.conj(t.fft.fft2(im2[im_slice]))
# Not sure if this is more or less stable than just the correlation
# maximum - requires some testing
cor = t.fft.ifft2(cor_fft / t.abs(cor_fft))
if im_np:
cor = cor.numpy()
return cor
def calc_frc(im1, im2, basis, im_slice=None, nbins=None, snr=1.):
"""Calculates a Fourier ring correlation between two images
This function requires an input of a basis to allow for FRC calculations
to be related to physical units.
Like other analysis functions, this can take input in numpy or pytorch,
and will return output in the respective format.
Parameters
----------
im1 : array
The first image, a complex or real valued array
im2 : array
The first image, a complex or real valued array
basis : array
The basis for the images, defined as is standard for datasets
im_slice : slice
Default is from 3/8 to 5/8 across the image, a slice to use in the processing.
nbins : int
Number of bins to break the FRC up into
snr : float
The signal to noise ratio (for the combined information in both images) to return a threshold curve for.
Returns
-------
freqs : array
The frequencies associated with each FRC value
FRC : array
The FRC values
threshold : array
The threshold curve for comparison
"""
im_np = False
if isinstance(im1, np.ndarray):
im1 = t.as_tensor(im1)
im_np = True
if isinstance(im2, np.ndarray):
im2 = t.as_tensor(im2)
im_np = True
if isinstance(basis, np.ndarray):
basis = t.tensor(basis)
if im_slice is None:
im_slice = np.s_[(im1.shape[0]//8)*3:(im1.shape[0]//8)*5,
(im1.shape[1]//8)*3:(im1.shape[1]//8)*5]
if nbins is None:
nbins = np.max(im1[im_slice].shape) // 4
cor_fft = t.fft.fftshift(t.fft.fft2(im1[im_slice]),dim=(-1,-2)) * \
t.fft.fftshift(t.conj(t.fft.fft2(im2[im_slice])),dim=(-1,-2))
F1 = t.abs(t.fft.fftshift(t.fft.fft2(im1[im_slice]),dim=(-1,-2)))**2
F2 = t.abs(t.fft.fftshift(t.fft.fft2(im2[im_slice]),dim=(-1,-2)))**2
di = np.linalg.norm(basis[:,0])
dj = np.linalg.norm(basis[:,1])
i_freqs = fftpack.fftshift(fftpack.fftfreq(cor_fft.shape[0],d=di))
j_freqs = fftpack.fftshift(fftpack.fftfreq(cor_fft.shape[1],d=dj))
Js,Is = np.meshgrid(j_freqs,i_freqs)
Rs = np.sqrt(Is**2+Js**2)
numerator, bins = np.histogram(Rs,bins=nbins,weights=cor_fft.numpy())
denominator_F1, bins = np.histogram(Rs,bins=nbins,weights=F1.detach().cpu().numpy())
denominator_F2, bins = np.histogram(Rs,bins=nbins,weights=F2.detach().cpu().numpy())
n_pix, bins = np.histogram(Rs,bins=nbins)
frc = np.abs(numerator / np.sqrt(denominator_F1*denominator_F2))
# This moves from combined-image SNR to single-image SNR
snr /= 2
threshold = (snr + (2 * snr + 1) / np.sqrt(n_pix)) / \
(1 + snr + (2 * np.sqrt(snr)) / np.sqrt(n_pix))
if not im_np:
bins = t.tensor(bins)
frc = t.tensor(frc)
threshold = t.tensor(threshold)
return bins[:-1], frc, threshold
def calc_vn_entropy(matrix):
"""Calculates the Von Neumann entropy of a density matrix
Will either accept a single matrix, or a stack of matrices. Matrices
are assumed to be Hermetian and positive definite, to be well-formed
density matrices
Parameters
----------
matrix : np.array
The nxn matrix or lxnxn stack of matrices to calculate the entropy of
Returns
-------
entropy: float or np.array
The entropy or entropies of the arrays
"""
if len(matrix.shape) == 3:
# Get the eigenvalues
eigs = [np.linalg.eigh(mat)[0] for mat in matrix]
# Normalize them to match standard density matrix form
eigs = [eig / np.sum(eig) for eig in eigs]
# And calculate the VN entropy!
entropies = [-np.sum(special.xlogy(eig,eig)) for eig in eigs]
return np.array(entropies)
else:
eig = np.linalg.eigh(matrix)[0]
entropy = -np.sum(special.xlogy(eig,eig))/np.sum(eig)
return entropy
def calc_top_mode_fraction(matrix):
"""Calculates the fraction of total power in the top mode of a density matrix
Will either accept a single matrix, or a stack of matrices. Matrices
are assumed to be Hermetian and positive definite, to be well-formed
density matrices
Parameters
----------
matrix : np.array
The nxn matrix or lxnxn stack of matrices to work from
Returns
-------
entropy: float or np.array
The fraction of power in the top mode of each matrix
"""
if len(matrix.shape) == 3:
# Get the eigenvalues
eigs = [np.linalg.eigh(mat)[0] for mat in matrix]
# Normalize them to match standard density matrix form
fractions = [np.max(eig) / np.sum(eig) for eig in eigs]
return np.array(fractions)
else:
eig = np.linalg.eigh(matrix)[0]
fraction = np.max(eig) / np.sum(eig)
return fraction
def calc_rms_error(field_1, field_2, align_phases=True, normalize=False,
dims=2):
"""Calculates the root-mean-squared error between two complex wavefields
The formal definition of this function is:
output = norm * sqrt(mean(abs(field_1 - gamma * field_2)**2))
Where norm is an optional normalization factor, and gamma is an
optional phase factor which is appropriate when the wavefields suffer
from a global phase degeneracy as is often the case in diffractive
imaging.
The normalization is defined as the square root of the total intensity
contained in field_1, which is appropriate when field_1 represents a
known ground truth:
norm = sqrt(mean(abs(field_1)**2))
The phase offset is an analytic expression for the phase offset which
will minimize the RMS error between the two wavefields:
gamma = exp(1j * angle(sum(field_1 * conj(field_2))))
This implementation is stable even in cases where field_1 and field_2
are completely orthogonal.
In the definitions above, the field_n are n-dimensional wavefields. The
dimensionality of the wavefields can be altered via the dims argument,
but the default is 2 for a 2D wavefield.
Parameters
----------
field_1 : array
The first complex-valued field
field_2 : array
The second complex-valued field
align_phases : bool
Default is True, whether to account for a global phase offset
normalize : bool
Default is False, whether to normalize to the intensity of field_1
dims : (int or tuple of python:ints)
Default is 2, the number of final dimensions to reduce over.
Returns
-------
rms_error : float or t.Tensor
The RMS error, or tensor of RMS errors, depending on the dim argument
"""
sumdims = tuple(d - dims for d in range(dims))
if align_phases:
# Keepdim allows us to broadcast the result correctly when we
# multiply by the fields
gamma = t.exp(1j * t.angle(t.sum(field_1 * t.conj(field_2), dim=sumdims,
keepdim=True)))
else:
gamma = 1
if normalize:
norm = 1 / t.mean(t.abs(field_1)**2, dim=sumdims)
else:
norm = 1
difference = field_1 - gamma * field_2
return t.sqrt(norm * t.mean(t.abs(difference)**2, dim=sumdims))
def calc_fidelity(fields_1, fields_2, dims=2):
"""Calculates the fidelity between two density matrices
The fidelity is a comparison metric between two density matrices
(i.e. mutual coherence functions) that extends the idea of the
overlap to incoherent light. As a reminder, the overlap between two
fields is:
overlap = abs(sum(field_1 * field_2))**2
Whereas the fidelity is defined as:
fidelity = trace(sqrt(sqrt(dm_1) <dot> dm_2 <dot> sqrt(dm_1)))**2
where dm_n refers to the density matrix encoded by fields_n such
that dm_n = fields_n <dot> fields_<n>.conjtranspose(), sqrt
refers to the matrix square root, and <dot> is the matrix product.
This is not a practical implementation, however, as it is not feasible
to explicitly construct the matrices dm_1 and dm_2 in memory. Therefore,
we take advantage of the alternate definition based directly on the
fields_<n> parameter:
fidelity = sum(svdvals(fields_1 <dot> fields_2.conjtranspose()))**2
In the definitions above, the fields_n are regarded as collections of
wavefields, where each wavefield is by default 2-dimensional. The
dimensionality of the wavefields can be altered via the dims argument,
but the fields_n arguments must always have at least one more dimension
than the dims argument. Any additional dimensions are treated as batch
dimensions.
Parameters
----------
fields_1 : array
The first set of complex-valued field modes
fields_2 : array
The second set of complex-valued field modes
dims : int
Default is 2, the number of final dimensions to reduce over.
Returns
-------
fidelity : float or t.Tensor
The fidelity, or tensor of fidelities, depending on the dim argument
"""
fields_1 = t.as_tensor(fields_1)
fields_2 = t.as_tensor(fields_2)
mult = fields_1.unsqueeze(-dims-2) * fields_2.unsqueeze(-dims-1).conj()
sumdims = tuple(d - dims for d in range(dims))
mat = t.sum(mult,dim=sumdims)
svdvals = t.linalg.svdvals(mat)
return t.sum(svdvals, dim=-1)**2
def calc_generalized_rms_error(fields_1, fields_2, normalize=False, dims=2):
"""Calculates a generalization of the root-mean-squared error between two complex wavefields
This function calculates an generalization of the RMS error which uses the
concept of fidelity to extend it to capture the error between
incoherent wavefields, defined as a mode decomposition. The extension has
several nice properties, in particular:
1) For coherent wavefields, it precisely matches the RMS error including
a correction for the global phase degeneracy (align_phases=True)
2) All mode decompositions of either field that correspond to the same
density matrix / mutual coherence function will produce the same
output
3) The error will only be zero when comparing mode decompositions that
correspond to the same density matrix.
4) Due to (2), one need not worry about the ordering of the modes,
properly orthogonalizing the modes, and it is even possible to
compare mode decompositions with different numbers of modes.
The formal definition of this function is:
output = norm * sqrt(mean(abs(fields_1)**2)
+ mean(abs(fields_2)**2)
- 2 * sqrt(fidelity(fields_1,fields_2)))
Where norm is an optional normalization factor, and the fidelity is
defined based on the mean, rather than the sum, to match the convention
for the root *mean* squared error.
The normalization is defined as the square root of the total intensity
contained in fields_1, which is appropriate when fields_1 represents a
known ground truth:
norm = sqrt(mean(abs(fields_1)**2))
In the definitions above, the fields_n are regarded as collections of
wavefields, where each wavefield is by default 2-dimensional. The
dimensionality of the wavefields can be altered via the dims argument,
but the fields_n arguments must always have at least one more dimension
than the dims argument. Any additional dimensions are treated as batch
dimensions.
Parameters
----------
fields_1 : array
The first set of complex-valued field modes
fields_2 : array
The second set of complex-valued field modes
normalize : bool
Default is False, whether to normalize to the intensity of fields_1
dims : (int or tuple of python:ints)
Default is 2, the number of final dimensions to reduce over.
Returns
-------
rms_error : float or t.Tensor
The generalized RMS error, or tensor of generalized RMS errors, depending on the dim argument
"""
fields_1 = t.as_tensor(fields_1)
fields_2 = t.as_tensor(fields_2)
npix = t.prod(t.as_tensor(fields_1.shape[-dims:],dtype=t.int32))
sumdims = tuple(d - dims - 1 for d in range(dims+1))
fields_1_intensity = t.sum(t.abs(fields_1)**2,dim=sumdims) / npix
fields_2_intensity = t.sum(t.abs(fields_2)**2,dim=sumdims) / npix
fidelity = calc_fidelity(fields_1, fields_2, dims=dims) / npix**2
result = fields_1_intensity + fields_2_intensity - 2 * t.sqrt(fidelity)
if normalize:
result /= fields_1_intensity
return t.sqrt(result)
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